51 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
and supposing x to be one of the roots of the equation B(oj) = 0, Baltzer 
predicates the n equations 
0 = [a m - A(x)} 4 ct m -i x + 4 ■ • ■ 
0 = {a m - A(x)}x 4 + 
0 = {a m - A{x)}x 2 4 
and the m equations 
0 = b n 4 b n _ x x + b n _ 2 x 2 4 
0 = b n x 4 b n _^t? 4 
0 = b n x 2 4 
and so deduces 
a m -A(x) a m _ x a m _ 2 
a m -A(x) a m _ x 
a m - A(x) . 
= 0 , 
which must thus be the equation in A(x) whose roots are A(/8 1 ), A (/3 2 ), ..., 
A (/3 n ). Since the coefficient of the highest power of A(x) in it is 
( — 1 ) n b 0 m } it follows that 
(-])*Jg‘.A0S 1 )A08 2 )...A (p n ) = 
a m 
«m-l 
V 2 • • • • 
CL m 
a m - .... 
a .... 
K- 1 
b n . 2 .... 
b n 
b n ~ 1 .... 
b n .... 
n+m ) 
as Hesse in 1858 had shown by direct transformation. 
The bigradient form of resultant is also used (§ 11, 7) to show that when 
A and B are of the same degree 
resultant (A, B4AA) = resultant (A, B). 
A fresh proof is given of Jacobi’s theorem * that if (p be a given 
function of the (m + n — l)* ;i degree in x, it is possible to determine two 
functions u , v of the (n — l) t}l , (m — l) th degrees so as to have 
wA 4 vB = S c/>, 
* Grelle’s Journ xv. (1835) p. 108, where however m=n. 
